The forced decoupling that we performed to obtain these expressions allows us to end up with results very similar to the second fundamental model of resonance proposed by [1]}. Indeed, the fixed points are given by the roots of the third degree polynomial in \(Z\)  (REF ), which has 1 or 3 real solutions depending on \(\delta \) , hence a bifurcation. The solutions of Eq. (REF ) are plotted in Fig. REF . For \(\delta <1\) , only one elliptic equilibrium exists, called the main branch, while for \(\delta \ge 1\) , two other fixed points appear, one of them being hyperbolic, hence the presence of separatrices in the phase space and the formal existence of a resonance. These results come from strong hypothesis and we see in Sect. REF  that the topology of the Hamiltonian (REF ) is different (see Table REF ). However, we show in Fig. REF  that these analytical expressions are accurate for small eccentricities.
